The theory of rotating incompressible fluids is an important part of hydrodynamics. In the in-viscous limit the linear analysis shows that all stationary solutions are pure two-dimensional (Taylor-Proudman theorem) and that a special kind of waves can propagate in this medium: inertial waves. In a reference frame rotating with the fluid this waves are plane waves whose frequency is 2 angular velocity times the cosine of the angle between the wave vector and the rotation axis.
In 1986 B. Bayly showed that two-dimensional stationary ?elliptic? flow is unstable with respect to excitation of three-dimensional inertial waves. The later studies have shown that it leads to the formation of ?weak wave turbulence?. In 2003 S. Galtier studied this problem using a helicity decomposition. However, the full theory of weak turbulence of inertial waves in rotating fluids is far from complete.
In this work [1] we present the Hamiltonian description of incompressible rotating fluid by using the Clebsch variables and study the effects of ?ellipticity?. We find the canonical transformation, which allows us to present the three-wave interaction Hamiltonian in normal variables in simple explicit form. We analyze the three-wave interaction amplitude and describe the turbulent cascade of the inertial waves. Finally we present the kinetic equation.

Authors thank the support of the Russian Science Foundation (Grant "Wave turbulence: theory, numerical simulation, experiment" No 14-22-00174).
[1]. Andrey A. Gelash, Victor S. L?vov and Vladimir E. Zakharov. Theory of weak turbulence of inertial waves in rotating fluids. In preparation.